Does image texture resolution impact rendering time?
Render time will likely be affected, but not much so, in theory it should make a negligible difference unless you are working close to your hardware limits. 

It will however have a significant impact in memory consumption. If you have little memory resources available for rendering this may have a more noticeable negative impact, like for example while GPU rendering on a card with low VRAM amount.

Texture size will significantly affect painting performance